Simple folding (folding along one line at a time) is a practical form of
origami used in manufacturing such as sheet metal bending. We prove strong
NP-completeness of deciding whether a crease pattern can be simply folded,
both for orthogonal paper with assigned orthogonal creases and for square
paper with assigned or unassigned creases at multiples of 45°. These
results settle a long standing open problem, where weak NP-hardness was
established for a subset of the models considered here, leaving open the
possibility of pseudopolynomial-time algorithms. We also formalize and
generalize the previously proposed simple folding models, and introduce new
infinite simple-fold models motivated by practical manufacturing. In the
infinite models, we extend our strong NP-hardness results, as well as
polynomial-time algorithms for rectangular paper with assigned or unassigned
orthogonal creases (map folding). These results motivate why rectangular maps
have orthogonal but not diagonal creases.